Now let \((M,g)\)  be any smooth spacetime, and let \(u\)  be a smooth future directed timelike vector field on \(M\)  normalized to \(\langle u,u\rangle = -1\) . For simplicity, assume \(\text{dim}(M) = 4\) . Define \(H = \frac{1}{3}\text{div}(u)\) . Letting \(\tau \)  denote the proper time of the flow lines of \(u\) , the Raychaudhuri equation [1]} gives
\(3\left(\frac{d H}{d\tau } + H^2\right)\,=\, -\text{Ric}(u,u) + 2\omega ^2 - 2\sigma ^2 + \text{div}(\nabla _u u),\) 
